Showing papers in "Pacific Journal of Mathematics in 1990"

Optimal paths for a car that goes both forwards and backwards.

Abstract: The path taken by a car with a given minimum turning radius has a lower bound on its radius of curvature at each point, but the path has cusps if the car shifts into or out of reverse gear. What is the shortest such path a car can travel between two points if its starting and ending directions are specified? One need consider only paths with at most 2 cusps or reversals

Matched pairs of lie groups associated to solutions of the yang-baxter equations

Abstract: Two groups G, H are said to be a matched pair if they act on each other and these actions, (a, /?), obey a certain compatibility condition In such a situation one may form a bicrossproduct group, denoted Gβ cχiQ H Also in this situation one may form a bicrossproduct Hopf, Hopf-von Neumann or Kac algebra obtained by simultaneous cross product and cross coproduct We show that every compact semi-simple simply-connected Lie group G is a member of a matched pair, denoted (G, G*)9 in a natural way As an example we construct the matched pair in detail in the case (SU(2), SU(2)*) where

Characterizations of conditional expectation-type operators.

Abstract: As is well-known, conditional expectation operators on various function spaces exhibit a number of remarkable properties related either to the underlying order structure of the given function space, or to the metric structure when the function space is equipped with a norm. Such operators are necessarily positive projections which are averaging in a precise sense to be described below and in certain normed function spaces are contractive for the given norm.

MULTIPLIERS OF H p AND BMOA

Abstract: [17] M. Mateljevic and M. Pavlovic, LP-behavior of the integral means of analyticfunctions, Studia Math., 77 (1983), 219-237.[18] M. Pavlovic, Mixed norm spaces of analytic and harmonic functions, I, Publ.Inst. Math. (Beograd), 40 (54), (1986), 117-141.[19] , An inequality for the integral means of Hadamard product, Proc. Amer.Math. Soc, 103 (1988), 404-406.[20] W. T. Sledd, On multipliers of H

About compressible viscous fluid flow in a bounded region

Abstract: This paper deals with the question of existence for all times of the solutions of a certain class of differential equations for small initial values, and with the asymptotic behavior of these solutions. This class of equations contains different models describing the flow of viscous compressible fluids, even under the influence of a magnetic field. 1. Introduction. We consider the initial-boundary value problem on a bounded domain Ωcl" with Dirichlet boundary conditions ( n) representing the relevant physical variables in their dependence on space and time. For the sake of simplicity we assume that Ωi c Rm+1 is a convex domain containing all physically reasonable values of X. The set might, e.g., include only positive values for density (which is usually the (ra +1) st component of X), and temperature. Then our equations have the form (E) X/ + /(X, VX) = LιxX + gι{x, t) (/ = 1, ... , m) ,

THE n-DIMENSIONAL ANALOGUE OF THE CATENARY: EXISTENCE AND NONEXISTENCE

Abstract: We study "heavy" A?-dimensional surfaces suspended from some prescribed (n — 1) -dimensional boundary data. This leads to a mean curvature type equation with a non-monotone right hand side. We show that the equation has no solution if the boundary data are too small, and, using a fixed point argument, that the problem always has a smooth solution for sufficiently large boundary data. Consider a material surface M of constant mass density which is suspended from an (n - 1 )-dimensional surface Γ in Rn x R+, R+ = {t > 0}, and hangs under its own weight. If M is given as graph of a regular function u: Ω —• R + on a domain Ω c Rw, n > 2, then u provides an equilibrium for the potential energy g7 under gravitational forces, = ί JΩ \Du\ Ω V Thus u solves the Dirichlet problem

On the Cohen-Macaulay property in commutative algebra and simplicial topology.

Abstract: A ring R is called a \"ring of sections\" provided R is the section ring of a sheaf (J/, X) of commutative rings defined over a base space X which is a finite partially ordered set given the order topology. Regard X as a finite abstract complex, where a chain in X corresponds to a simplex. In specific instances of (s/ ,X), certain algebraic invariants of R are equivalent to certain topological invariants of X.

Weierstrass points on Gorenstein curves

Abstract: Let Y denote a smooth, projective curve of genus g defined over C. A point P G Y is a Weierstrass point if there exists a rational function on Y with a pole only at P of order at most g, or if there exists a regular differential on Y which vanishes at P to order at least g, or if the divisor gP is special. However, the most "functorial" way to define Weierstrass points is as the zeros of the wronskian, a section of the (g(g + l)/2)th tensor power of the canonical bundle on Y. It is this last definition that we use as the foundation for defining Weierstrass points on singular curves. What is essential is that the sheaf of dualizing differentials should be locally free and this is exactly the property satisfied by Gorenstein curves. Let X be an integral, projective Gorenstein curve of arithmetic genus g > 0 over C. Let ω denote the bundle of dualizing differentials on X and let Sf denote an invertible sheaf on X. Put s = dim H°(X,J?) = h°(&). Assume s > 0 and choose a basis φΪ9...9φsfoτH0(X,5?). We will define a section of S?®sΘω^s'1^2 as follows: Suppose that {U^} is a covering of X by open subsets such that ^f(U^) and ω(UM) are free ^(^ (α) )-modules generated by ψ(a) and τ^a\ respectively. Define FJ f e YilJ^^x) inductively

Compactification of $M_{{\bf P}_3}(0,2)$ and Poncelet pairs of conics.

Abstract: Let M(0, 2) denote the quasi-projective variety of isomorphism classes of stable rank 2 vector bundles on P3(C) with C1=0 and C2=2 . In this paper we study a natural (irreducible) compactification of M(0, 2) and describe explicitly the sheaves on P3 which occur in the closure of M(0, 2) in the moduli space of semi-stable sheaves on P3 with c1= 0, c2=2 and c3=0.

Seifert surfaces of knots in s4

Abstract: This paper uses some ideas from 3-dimensional topology to study knots in S4 . We show that the Poincare conjecture implies the existence of a non-fibered knot whose complement fibers homotopically. In a different direction, we show that Gromov's norm is an obstruction to a knot having a Seifert surface made out of Seifert fibered spaces, and hence to being ribbon. We also prove that any 3-manifold is invertibly homology cobordant to a hyperbolic 3-manifold, so that every knot in S4 has a hyperbolic Seifert surface. One of the reasons that the study of knots in the 4-sphere has a special character is that the Seifert surfaces that such knots bound are 3-dimensional. Hence the peculiar nature of the topology of 3manifolds can lead to interesting behavior of 2-knots. In this paper we give several examples of this principle. The first example is to show that the 3-dimensional Poincare conjecture implies the existence of non-fibered (topological) knots in *S 4 whose exteriors are homotopy equivalent to the exterior of a fibered knot. (Similar phenomena have been noticed by J. Hillman and C. B. Thomas [12, 13] and S. Weinberger [32].) The second instance is to see how the existence of a "geometric structure" on a Seifert surface influences topological properties of the knot. Restrictions on the possible geometric structures are obtained via "Gromov's norm" of a 2-knot, defined below. We show that a knot with non-zero norm cannot have a Seifert surface which is a connected sum of Seifert-fiber ed 3-manifolds. In particular, the norm is seen to be an obstruction to a knot in S4 being ribbon. A similar obstruction has been found by Bruce Trace [30]. In contrast, we will show that any knot has a Seifert surface which is a hyperbolic manifold. This follows from Theorem 2.6, which states that any 3-manifold has an invertible homology cobordism to a hyperbolic manifold. A cobordism W from M to N is called invertible if there is another cobordism W, so that W U N W = M x I. Without the requirement that the homology cobordism be invertible, this theorem is due to R. Myers [23].